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On Channel Assignment Of Graphs. Author : Hsin-Ju Wu Adviser : Yung-Ling Lai Speaker : Shr-Jia Hung. Outline. Motivation and Definition Off-line Labeling On-line Labeling Conclusion and Future work. Outline. Motivation and Definition Off-line Labeling
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On Channel Assignment Of Graphs Author : Hsin-Ju Wu Adviser : Yung-Ling Lai Speaker : Shr-Jia Hung
Outline • Motivation and Definition • Off-line Labeling • On-line Labeling • Conclusion and Future work
Outline • Motivation and Definition • Off-line Labeling • On-line Labeling • Conclusion and Future work
Motivation • Channel assignment problem • the number of finite frequencies • Use a graph to model it.
Motivation • Vertices transmitters • Edges Distances Adjacent very close Distance 2 Close B C A
Definition • k-L(p,q) labeling f for a given graph G=(V,E), is a function f : V→{ 0,1,…,k } such that | f(x)-f(y) | p if d(x,y)=1 and | f(x)-f(y) | q if d(x,y)=2. • The L(p,q) labeling number of graph G is then defined as:
Outline • Motivation and Definition • Off-line Labeling • On-line Labeling • Conclusion and Future work
Off-line Labeling • L(d,1) labeling on • L(d,1) labeling on
Outline • Motivation and Definition • Off-line Labeling • On-line Labeling • Conclusion and Future work
Definition of online labeling • Given a graph G. • The vertices are given one-by-one arbitrarily. • Only the adjacency relation between the given vertices are known. • Satisfy the condition of L(2,1)-labeling. • Give a label right away which is not changeable later.
Online L(2,1) labeling of path • Path algorithm Call Function Get_available_Number(N1,N2)
Get_available_Number(N1,N2) Get_available_Number(N1,N2) N1=0 N1=1 N1=2 N2=0 N2=0 N2=1 N2=0 N2=1 N2=2
N1=1 and N2=0 L1 x
N1=1 and N2=1 vj L1 x L1 x
L2 x L1 L2 L2 x x L1 L1 N1=2 and N2=0,1,2
Time Complexity • Path algorithm 3.1 O(n) Call Function Get_available_Number(N1,N2)
Time Complexity • Path algorithm Get_available_Number(N1,N2) O(n)
Time Complexity • Path algorithm 3.1 O(n2) Call Function Get_available_Number(N1,N2)
Online L(2,1) labeling of path Pattern A 26
Online L(2,1) labeling of cycle Pattern B 28
Star S9 7 9 8 6 0 2 7 3 5 1 9 0 2 4 4 6 5 3
Online L(2,1) labeling of star n-1 1 n 0 1 2
Online L(2,1) labeling of star 3 2 2 1 n 4 0 1
Online---Other graph bound Double star: Full binary tree: 33
Outline • Motivation and Definition • Off-line Labeling • On-line Labeling • Conclusion and Future work
Conclusion • Off-line L(d,1) labeling - - • On-line L(2,1) labeling - Path - Cycle - Star - Double star - Full binary tree
Future Work • On-line L(2,1) labeling • K2xPn , K2xCn • Wheel • Complete bipartite graph • relation with max degree • relation with radius , diamter • relation with density(size/order)